Question

classify the critical point \\((0, 0)\\) of the given linear system by computing the trace \\(\tau\\), determinant \\(\delta\\), and discriminant and using the figure.
\\
\

$$\begin{aligned} x &= -7x + 5y \\\\ y &= 3x - 9y \\end{aligned}$$

\\
trace \\(\tau =\\)
determinant \\(\delta =\\)
discriminant \\(\tau^2 - 4\delta =\\)
classify the critical point \\((0, 0)\\).

• degenerate unstable node
• unstable spiral
• stable spiral
• stable node
• unstable node
• center
• saddle
• degenerate stable node

Explanation:

Construct the coefficient matrix

$$ A =
LATEXBLOCK0
$$

Compute trace, determinant, and discriminant

$$
LATEXBLOCK1
$$

Classify the critical point

$$
LATEXBLOCK2
$$

Since \(\Delta > 0\) and \(\tau^2 - 4\Delta > 0\), the eigenvalues are real, distinct, and have the same sign. Because \(\tau < 0\), both eigenvalues are negative, which classifies the critical point \((0,0)\) as a stable node.

Answer:

Question 1

trace \(\tau =\) <blank>-16</blank>

Question 2

determinant \(\Delta =\) <blank>48</blank>

Question 3

discriminant \(\tau^2 - 4\Delta =\) <blank>64</blank>

Question 4

Classify the critical point \((0, 0)\).

• degenerate unstable node
• unstable spiral
• stable spiral
• stable node (Correct answer)
• unstable node
• center
• saddle
• degenerate stable node